Let X be a smooth projective geometrically connected curve over
a finite field with function field K. Let cal G be a connected semisimple group
scheme over X. Under certain hypotheses we prove the equality of
two numbers associated with cal G.
The first is an arithmetic invariant, its Tamagawa number. The second
is a geometric invariant, the number of connected components of the moduli
stack of $\cal G-torsors on X. Our results are most useful for studying
connected components as much is known about Tamagawa numbers.

Let $X$ be a smooth projective geometrically connected curve over
a finite field with function field $K$. Let $\G$ be a connected semisimple group
scheme over $X$. Under certain hypotheses we prove the equality of
two numbers associated with $\G$.
The first is an arithmetic invariant, its Tamagawa number. The second
is a geometric invariant, the number of connected components of the moduli
stack of $\G$-torsors on $X$. Our results are most useful for studying
connected components as much is known about Tamagawa numbers.

In this paper we prove that a generalized version of the Minimal
Resolution Conjecture given by Mustaja holds for certain
general sets of points on a smooth cubic surface X subset
{mathbb P}3. The main tool used is Gorenstein liaison theory and, more
precisely, the relationship between the free resolutions of two linked schemes.

In this paper we prove that a generalized version of the Minimal
Resolution Conjecture given by Musta\c{t}\v{a} holds for certain
general sets of points on a smooth cubic surface $X \subset
\PP^3$. The main tool used is Gorenstein liaison theory and, more
precisely, the relationship between the free resolutions of two linked schemes.

Given a positive continuous function mu on the
interval 0 < t leq 1, we consider the space of so-called mu-Bloch
functions on the unit ball. If mu(t) = t, these are the classical
Bloch functions. For mu, we define a metric Fzmu(u) in
terms of which we give a characterization of mu-Bloch functions.
Then, necessary and sufficient conditions are obtained in order that
a composition operator be a bounded or compact operator between
these generalized Bloch spaces. Our results extend those of Zhang
and Xiao.

Given a positive continuous function $\mu$ on the
interval $0<t\le1$, we consider the space of so-called $\mu$-Bloch
functions on the unit ball. If $\mu(t)=t$, these are the classical
Bloch functions. For $\mu$, we define a metric $F_z^\mu(u)$ in
terms of which we give a characterization of $\mu$-Bloch functions.
Then, necessary and sufficient conditions are obtained in order that
a composition operator be a bounded or compact operator between
these generalized Bloch spaces. Our results extend those of Zhang
and Xiao.

Let R be a homomorphic image of a Gorenstein local ring. Recent
work has shown that there is a bridge between Auslander categories
and modules of finite Gorenstein homological dimensions over R.
We use Gorenstein dimensions to prove new results about Auslander
categories and vice versa. For example, we establish base change
relations between the Auslander categories of the source and target
rings of a homomorphism varphi : R \to S of finite flat dimension.

Let $R$ be a homomorphic image of a Gorenstein local ring. Recent
work has shown that there is a bridge between Auslander categories
and modules of finite Gorenstein homological dimensions over $R$.

We use Gorenstein dimensions to prove new results about Auslander
categories and vice versa. For example, we establish base change
relations between the Auslander categories of the source and target
rings of a homomorphism $\varphi \colon R \to S$ of finite flat dimension.

We produce ample (resp. NEF, eventually free) divisors in the
Kontsevich space overline{cal M}0,n (mathbb Pr, d) of n-pointed,
genus 0, stable maps to mathbb Pr, given such divisors in
overline{cal M}0,n+d. We prove that this produces all ample (resp. NEF,
eventually free) divisors in overline{cal M}0,n (mathbb Pr,d).
As a consequence, we construct a contraction of the boundary
bigcupk=1lfloor d/2 \rfloor Deltak,d-k in
overline{cal M}0,0(\mathbb Pr,d), analogous to a contraction of
the boundary bigcupk=3lfloor n/2 rfloor
\tilde{Delta}k,n-k in overline{cal M}0,n first constructed by Keel
and McKernan.

We produce ample (resp.\ NEF, eventually free) divisors in the
Kontsevich space $\Kgnb{0,n} (\mathbb P^r, d)$ of $n$-pointed,
genus $0$, stable maps to $\mathbb P^r$, given such divisors in
$\Kgnb{0,n+d}$. We prove that this produces all ample (resp.\ NEF,
eventually free) divisors in $\Kgnb{0,n}(\mathbb P^r,d)$.
As a consequence, we construct a contraction of the boundary
$\bigcup_{k=1}^{\lfloor d/2 \rfloor} \Delta_{k,d-k}$ in
$\Kgnb{0,0}(\mathbb P^r,d)$, analogous to a contraction of
the boundary $\bigcup_{k=3}^{\lfloor n/2 \rfloor}
\tilde{\Delta}_{k,n-k}$ in $\kgnb{0,n}$ first constructed by Keel
and McKernan.

The space now known as complete Erdös
space\cerdos was introduced by Paul Erdös in 1940 as the
closed subspace of the Hilbert space ell2 consisting of all
vectors such that every coordinate is in the convergent sequence
{0} \cup {1/n : n \in {mathbb N}}. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of \cerdos.
As an application we determine the class
of factors of \cerdos. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces ellp according to the `Erdös method' are homeomorphic
to \cerdos. A novel application states that if I is a
Polishable Fsigma-ideal on $\omega$, then I with the Polish
topology is homeomorphic to either {mathbb Z}, the Cantor set 2omega,
{mathbb Z} \times 2omega, or \cerdos. This last result answers a
question that was asked
by Stevo Todorcevic.

The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.

Let p be a prime, and let f \from {mathbb Z}/p {mathbb Z} \rightarrow
{mathbb R} be a function with {mathbb E} f = 0 and || \widehat{f}
||1 leq 1. Then
minx \in {mathbb Z}/p{mathbb Z} |f(x)| = O(log p){-1/3 + epsilon}.
One should think of f as being "approximately continuous"; our
result is then an "approximate intermediate value theorem".
As an immediate consequence we show that if A \subseteq {mathbb Z}/p{mathbb Z} is a
set of cardinality lfloor p/2 rfloor, then
sumr |\widehat{1A}(r)| > > (log p){1/3 - epsilon}. This
gives a result on a "mod p" analogue of Littlewood's well-known
problem concerning the smallest possible L1-norm of the Fourier
transform of a set of n integers.
Another application is to answer a question of Gowers. If A
\subseteq {mathbb Z}/p{mathbb Z} is a set of size lfloor p/2 rfloor, then there is
some x \in {mathbb Z}/p{mathbb Z} such that
||A \cap (A + x)| - p/4 | = o(p).

Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow
\mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f}
\Vert_1 \leq 1$. Then
$\min_{x \in \Zp} |f(x)| = O(\log p)^{-1/3 + \epsilon}$.
One should think of $f$ as being ``approximately continuous''; our
result is then an ``approximate intermediate value theorem''.
As an immediate consequence we show that if $A \subseteq \Zp$ is a
set of cardinality $\lfloor p/2\rfloor$, then
$\sum_r |\widehat{1_A}(r)| \gg (\log p)^{1/3 - \epsilon}$. This
gives a result on a ``mod $p$'' analogue of Littlewood's well-known
problem concerning the smallest possible $L^1$-norm of the Fourier
transform of a set of $n$ integers.

Another application is to answer a question of Gowers. If $A
\subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is
some $x \in \Zp$ such that
\[ | |A \cap (A + x)| - p/4 | = o(p).\]

Let Theta = (alpha,beta) be a point in {mathbb R}2, with 1,alpha,
beta linearly independent over {mathbb Q}. We attach to Theta
quadruple Omega(Theta) of exponents that measure the quality
of approximation to Theta both by rational points and by
rational lines. The two "uniform" components of Omega(Theta)
are related by an equation due to Jarnik, and the four
exponents satisfy two inequalities that refine Khintchine's
transference principle. Conversely, we show that for any quadruple
Omega fulfilling these necessary conditions, there exists
a point Theta \in {mathbb R}2 for which Omega(Theta) = Omega.

Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,
\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a
quadruple $\Omega(\Theta)$ of exponents that measure the quality
of approximation to $\Theta$ both by rational points and by
rational lines. The two ``uniform'' components of $\Omega(\Theta)$
are related by an equation due to Jarn\'\i k, and the four
exponents satisfy two inequalities that refine Khintchine's
transference principle. Conversely, we show that for any quadruple
$\Omega$ fulfilling these necessary conditions, there exists
a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.

We consider the problem of determining for which square integrable
functions f and g on the polydisk the densely defined Hankel
product HfHg\ast is bounded on the Bergman space of the
polydisk. Furthermore, we obtain similar results for the mixed
Haplitz products HgT\bar{f} and TfHg*, where f
and g are square integrable on the polydisk and f is analytic.

We consider the problem of determining for which square integrable
functions $f$ and $g$ on the polydisk the densely defined Hankel
product $H_{f}H_g^\ast$ is bounded on the Bergman space of the
polydisk. Furthermore, we obtain similar results for the mixed
Haplitz products $H_{g}T_{\bar{f}}$ and $T_{f}H_{g}^{*}$, where $f$
and $g$ are square integrable on the polydisk and $f$ is analytic.

Natural sufficient conditions for a polynomial to have a local minimum
at a point are considered. These conditions tend to hold with
probability 1. It is shown that polynomials satisfying these
conditions at each minimum point have nice presentations in terms of
sums of squares. Applications are given to optimization on a compact
set and also to global optimization. In many cases, there are degree
bounds for such presentations. These bounds are of theoretical
interest, but they appear to be too large to be of much practical use
at present. In the final section, other more concrete degree bounds
are obtained which ensure at least that the feasible set of solutions
is not empty.

Natural sufficient conditions for a polynomial to have a local minimum
at a point are considered. These conditions tend to hold with
probability $1$. It is shown that polynomials satisfying these
conditions at each minimum point have nice presentations in terms of
sums of squares. Applications are given to optimization on a compact
set and also to global optimization. In many cases, there are degree
bounds for such presentations. These bounds are of theoretical
interest, but they appear to be too large to be of much practical use
at present. In the final section, other more concrete degree bounds
are obtained which ensure at least that the feasible set of solutions
is not empty.

This paper shows the existence and uniqueness of Klyachko models for
irreducible unitary representations of GL5(cal F), where cal F is
a p-adic field. It is an extension of the work of Heumos and Rallis on GL4(cal F).

This paper shows the existence and uniqueness of Klyachko models for
irreducible unitary representations of $\GL_5(\CF)$, where $\CF$ is
a $p$-adic field. It is an extension of the work of Heumos and Rallis on $\GL_4(\CF)$.